Operator overview

In migration by downward-continuation, the wavefield at depth 127#127 is obtained by phase-shift from the wavefield at depth z ()  

 130#130 (273)

where the depth wavenumber k_z depends linearly through a Taylor series expansion on its value in the reference medium (k_zo) and the slowness difference in the depth interval from z to 127#127, 682#682: 

 683#683 (274)

where, by definition, 684#684, and 244#244 denotes spatial position at depth z. The expression for 685#685 can take many different forms, summarized in ().

From Equations () and (), we can write that

   686#686 (275)

Equation () represents a general form of the main mixed-domain downward-continuation operator. This operator can be broken up into a group of functional operators as follows:

• Wavefield continuation using mixed-domain phase-shift:

     687#687 (276)

• Imaging condition which transforms the wavefield into an image at any given depth level:

     688#688 (277)

128#128 and 129#129 are the wavefields at depths 127#127 and z respectively, 175#175 is the depth step, s_o is the constant reference slowness in the slab from z to 175#175,689#689 is the variable slowness in the same depth slab, and 690#690 represents the depth wavenumber expressed using the double-square root equation, and which is a function of the reference slowness (s_o).

For Equations () and (), we can distinguish 5 functional operators. Each operator is initialized with a call to a function (XXin) and executed with a call to another function (XXop). In a typical example, the functional operators perform the following tasks:

1.

Wavefield continuation operator (WCin & WCop)

Continues the wavefield between two depth levels, using one or more reference slownesses.

Interface: integer function WCop(wfld,iws,izs,ith,FKop,FXop) result(st)

• complex, dimension(:,:,:,:,:), pointer :: wfld (wavefield slice)
• integer :: iws (index of the frequency slice)
• integer :: izs (index of the depth slice)
• integer :: ith (thread number)
• FKop :: f-k operator
• FXop :: f-x operator

Implemented examples:

• weimwc1 (mixed-domain wavefield continuation operator for 1 reference slowness)
• weimwcN mixed-domain wavefield continuation operator for N reference slownesses)

2.

Slowness operator (SLin & SLop)

Selects the number and values of the reference slownesses (s_o), and sets-up the interpolation map between the wavefields continued using the various reference slownesses.

Interface: integer function SLop() result(st)

Implemented examples:

• weislo1 (slowness selector for 1 reference slowness)
• weisloN (slowness selector for N reference slownesses)

3.

f-k operator (FKin & FKop)

Performs phase-shift using the full 3-D DSR equation (), the common-azimuth equation (), or the offset plane-waves equation ().

Interface: integer function FXop(iws,izs,ifk,ith,wfld) result(st)

• integer :: iws (index of the frequency slice)
• integer :: izs (index of the depth slice)
• integer :: ifk (index of the reference velocity)
• integer :: ith (thread number)
• complex, dimension(:,:,:,:,:), pointer :: wfld (wavefield slice)

Implemented examples:

• weiwem (3-D prestack or 3-D offset plane-waves phase-shift)
• weicam (3-D common-azimuth phase-shift)

4.

f-x operator (FXin & FXop)

Performs phase shift that accounts for lateral slowness variation. Examples of (f-x) operators include but are not limited to split-step Fourier (), local Born Fourier or local Rytov Fourier (), Fourier Finite-Difference (), generalized screen propagators (), etc. Interface: integer function FXop(iws,izs,ifk,ith,wfld) result(st)

• integer :: iws (index of the frequency slice)
• integer :: izs (index of the depth slice)
• integer :: ifk (index of the reference velocity)
• integer :: ith (thread number)
• complex, dimension(:,:,:,:,:), pointer :: wfld (wavefield slice)

Implemented example:

• weissf (Split-step correction)

5.

Imaging operator (IGin & IGop)

Performs imaging in the offset-domain or the offset ray-parameter domain. This operator can also incorporate amplitude-preserving corrections.

Interface: integer function IGop(wfld,iws,ith) result(st)

• complex, dimension(:,:,:,:,:), pointer :: wfld (wavefield slice)
• integer :: iws (index of the frequency slice)
• integer :: ith (thread number)

Implemented examples:

• weihcig (Offset-domain common image-gathers)
• weipcig (Offset ray-parameter p_h common image-gathers)